---
title: "Aliasing; Nyquist of 5cos1500πt − …"
subject: "Signals and Systems"
chapter: 4
key: "semester_4/signals_and_systems:aliasing-nyquist-three-tones"
source: https://www.bctengineer.com/q/semester_4/signals_and_systems/aliasing-nyquist-three-tones
site: BCT Engineering
---

# Aliasing; Nyquist of 5cos1500πt − …

Signals and Systems · Chapter 4: Sampling

- Asked in: 2078 Chaitra, 2+4 marks
- Video: [Sampling, Aliasing & Nyquist Theorem](https://www.youtube.com/watch?v=yWqrx08UeUs) — 0612 TV w/ NERDfirst
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/signals_and_systems/aliasing-nyquist-three-tones
- Whole chapter: https://www.bctengineer.com/md/semester_4/signals_and_systems/chapter-4

## Question

What do you mean by aliasing? Compute Nyquist rate for x(t) = 5cos(1500πt) − 2sin(2000πt) + 3sin(1850πt).

## Short answer

Aliasing is distortion when sampling below the Nyquist rate causes spectral overlap that folds high frequencies into lower ones. For x(t)=5cos(1500πt)−2sin(2000πt)+3sin(1850πt), frequencies are 750, 1000, and 925 Hz; maximum is 1000 Hz, so Nyquist rate is 2000 Hz.

## Answer

#### Aliasing
**Aliasing** is the distortion that occurs when a signal is sampled at a rate lower than its Nyquist rate ($f_s < 2f_M$). The shifted copies of the spectrum in the sampled signal overlap, so a high-frequency component "folds back" and appears as a lower frequency (its *alias*). After overlap the original spectrum cannot be separated by any filter, so the signal cannot be recovered exactly.

Example: a 700 Hz tone sampled at $f_s = 1000$ Hz gives the same samples as a $1000-700 = 300$ Hz tone.

```text
   overlap (aliasing) when ws < 2wM
     ___       ___       ___
    /   \     /   \     /   \
   /     \   /     \   /     \
--/-------\-/-------\-/-------\--> w
       -ws  \ /  0   \ /  ws
             X        X   <- overlapped region
```

#### Nyquist rate of $x(t) = 5\cos(1500\pi t) - 2\sin(2000\pi t) + 3\sin(1850\pi t)$

| Term | $\omega$ (rad/s) | $f = \omega/2\pi$ (Hz) |
|---|---|---|
| $5\cos(1500\pi t)$ | $1500\pi$ | $750$ |
| $-2\sin(2000\pi t)$ | $2000\pi$ | $1000$ |
| $3\sin(1850\pi t)$ | $1850\pi$ | $925$ |

The maximum frequency is $f_M = 1000$ Hz.
$$
f_N = 2 f_M = 2 \times 1000 = 2000\ \text{Hz}
$$
**Answer: Nyquist rate = 2000 Hz (2000 samples/s, i.e. $\omega_N = 4000\pi$ rad/s).**

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. Aliasing happens when
   A. fs > 2fM
   B. The signal is not periodic
   C. fs < 2fM so spectral copies overlap
   D. The filter has gain > 1
   Answer: C. High frequencies fold back. (Key point: Aliasing)

2. 700 Hz sampled at 1000 Hz gives samples equal to a tone at
   A. 700 Hz
   B. 1700 Hz
   C. 500 Hz
   D. 300 Hz
   Answer: D. 1000 - 700 = 300. (Key point: Example)

3. 3sin(1850πt) has frequency
   A. 925 Hz
   B. 1850 Hz
   C. 1000 Hz
   D. 3700 Hz
   Answer: A. 1850π/2π = 925. (Key point: Frequency of 1850π)

4. Nyquist rate of 5cos1500πt - 2sin2000πt + 3sin1850πt
   A. 1850 Hz
   B. 2000 Hz
   C. 1500 Hz
   D. 4000 Hz
   Answer: B. fM = 1000 Hz from the 2000π term. (Key point: Nyquist rate)

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